Determine the quadratic function, f(x), with x intercepts at x=7 and x=-2 and y intercept at y=5.
I have been struggling with this question for days. I have phoned my teacher and gotten her help, but I still can't figure it out. If someone could please help me with this, that would be great. Thanks
For days ????
since the x-intercepts of a function are the same as the roots of the corresponding equation
x=7 ----> a factor of (x-7)
x=-2 ---> a factor of (x+2)
so f(x) = a(x-7)(x+2)
we are told that (0,5) lies on it, so
5 = a(-7)(2)
a = -5/14
f(x) = (-5/14)(x-7)(x+2)
or
f(x) = (-5/14)x^2 - (25/14)x + 5
check:
https://www.wolframalpha.com/input/?i=solve+(-5%2F14)x%5E2+-+(25%2F14)x+%2B+5%3D0
To determine the quadratic function, f(x), with x-intercepts at x=7 and x=-2, and a y-intercept at y=5, we can use the factored form of a quadratic function. The factored form is written as:
f(x) = a(x - r)(x - s)
Where "r" and "s" are the roots (x-intercepts) of the quadratic function, and "a" is a constant.
Given that the x-intercepts are at x=7 and x=-2, we can substitute those values into the equation:
f(x) = a(x - 7)(x - (-2))
= a(x - 7)(x + 2)
Now, to find the value of "a," we can use the y-intercept at y=5. The y-intercept represents the point (0, y) on the graph. In this case, the y-intercept is (0, 5). We can substitute these values into the equation:
5 = a(0 - 7)(0 + 2)
= a(-7)(2)
= -14a
To solve for "a," divide both sides of the equation by -14:
5 / -14 = a
-5/14 = a
So, the value of "a" is -5/14. Now we can write the quadratic function, f(x), with the given x and y-intercepts:
f(x) = (-5/14)(x - 7)(x + 2)
Therefore, the quadratic function with x-intercepts at x=7 and x=-2, and a y-intercept at y=5 is f(x) = (-5/14)(x - 7)(x + 2).
To determine the quadratic function, f(x), with the given x-intercepts and y-intercept, we can use the fact that for a quadratic function in the form f(x) = ax^2 + bx + c, the x-intercepts occur when f(x) equals zero, and the y-intercept occurs when x equals zero.
First, let's find the equation for the x-intercepts. The x-intercepts are given as x = 7 and x = -2. Since f(x) equals zero at these x-values, we can set up the following equations:
When x = 7: f(7) = a(7)^2 + b(7) + c = 0
When x = -2: f(-2) = a(-2)^2 + b(-2) + c = 0
Simplifying these equations, we have:
49a + 7b + c = 0 -- Equation 1
4a - 2b + c = 0 -- Equation 2
Next, let's find the equation for the y-intercept. The y-intercept is given as y = 5. Since the y-intercept occurs when x equals zero, we can set up the following equation:
When x = 0: f(0) = a(0)^2 + b(0) + c = 5
Simplifying this equation, we have:
c = 5 -- Equation 3
Now we have a system of three equations (Equations 1, 2, and 3) with three variables (a, b, and c). We can solve this system to determine the quadratic function, f(x).
Substitute Equation 3 into Equations 1 and 2:
49a + 7b + 5 = 0 -- Equation 4
4a - 2b + 5 = 0 -- Equation 5
Now solve the system of Equations 4 and 5:
Multiply Equation 5 by 7:
28a - 14b + 35 = 0 -- Equation 6
Subtract Equation 6 from Equation 4:
21a + 21 = 0 -- Equation 7
Solve Equation 7:
21a = -21
a = -1
Substitute the value of a back into Equation 4:
-49 + 7b + 5 = 0
7b - 44 = 0
7b = 44
b = 44/7
b = 6.2857 (rounded to four decimal places)
Substitute the values of a and b into Equation 3 to find c:
c = 5
Therefore, the quadratic function, f(x), with x-intercepts at x = 7 and x = -2 and y-intercept at y = 5 is:
f(x) = -x^2 + 6.2857x + 5